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TRANSVERSE KICK ANALYSIS OF SSR1 DUE TO POSSIBLE
GEOMETRICAL VARIATIONS IN FABRICATION*
M. Awida#, I. Gonin, P. Berutti, T. Khabiboulline, and V. Yakovlev
Fermilab, Batavia, IL 60510, USA
Abstract Due to fabrication tolerance, it is expected that some
geometrical variations could happen to the SSR1 cavities
of Project X, like small shifts in the transverse direction
of the beam pipe or the spoke. It is necessary to evaluate
the resultant transverse kick due to these geometrical
variations, in order to make sure that they are within the
limits of the correctors in the solenoids. In this paper, we
report the transverse kick values for various fabrications
errors and the sensitivity of the beam to these errors.
INTRODUCTION
Spoke cavities as a low β accelerating cavities plays a
major role in the continuous wave linac of Project X.
Project X is conceived as the next generation
superconducting linac to be built in Fermilab targeting the
intensity frontier with focus on the study of rare
subatomic processes and supporting neutrino experiments
[1-2]. Two kinds of spoke cavities namely SSR1 and
SSR2 will be used to accelerate the proton beam at
relative velocity of β=0.22 and β=0.47, respectively.
SSR1 cavities are currently under production for the
Project X injection experiment (PXIE) which is planned
to test the integrated systems of Project X front end [3-4].
PXIE consists of an ion source capable of delivering 5
mA (nominal) at 30 keV followed by a LEBT section, a 5
mA RFQ, a MEBT section with integrated wideband
chopper, as shown in Figure 1. Two superconducting
crymodules are then used to accelerate the beam from 2.1
MeV at the end of the MEBT section to 40 MeV. The two
crymodules are one of seven half wave resonators
(β=0.11) and the other is of eight spoke cavities (SSR1,
β=0.22) [5].
Due to the sensitive nature of CW linacs to losses and
activation, stringent requirements are imposed on the
cavities design, specially the losses of higher order modes
and the beam kicks. In case of the spoke cavities,
transverse beam kicks could happen due to possible
geometrical variations that might happen due to
fabrication tolerances, which include:-
Beam pipe shift in transverse direction.
Spoke shift in transverse direction.
Spoke shift in longitudinal direction.
It is imperative to study the effect of these geometrical
variations on the performance of the cavity especially
from transverse kick point of view. In this paper, we
report the transverse kick values for SSR1 due to various
fabrications’ geometrical variations and the sensitivity of
the beam to these errors.
SSR1
Figure 2 shows a quarter of SSR1’s RF domain. The
cavity gap distance was designed to accelerate particle at
relative velocity β=0.22. Cavity operates at 325 MHz
(sub-harmonic of 1.3 GHz) with bandwidth of 90 Hz. The
nominal gain per cavity is 2 MeV with projected
maximum magnetic field of 60 mT and max surface
electric field of 39 MV/m.
Figure 2 depicts the possible geometrical variations that
could induce transverse beam kicks. PXIE lattice includes
correctors on the focusing solenoid in between cavities (c-
s-c-c-s-c-c-s-c-c-s-c) to correct for such beam kicks. Each
solenoid has two correctors; one for each transverse
direction that can correct for up to 10 mrad angular beam
deviation [6]. Therefore, it is imperative to investigate the
possible beam transverse kicks and make sure that they
are well below this limit.
Figure 2: SSR1 geometry and the possible geometrical
variations.
Figure 1: PXIE Layout.
________________________________________________________________________________________________________
*Work supported by the US Department of Energy #mhassan@fnal.gov
FERMILAB-CONF-12-159-TD
Operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the United States Department of Energy
KICK ANALYSIS
In order to analyse the transverse kick that could
happen to the beam due to certain geometrical variations,
we have built a simulation model for SSR1 in Comsol
Multiphysics [7]. The three kinds of geometrical
variations have been considered namely; displacements in
the spoke location with respect to the symmetry plane of
the cavity both in the transverse and longitudinal
directions of up to 1 mm and displacement in the beam
pipe location in the transverse direction of up to 1 mm.
Four cases of these geometrical variations, as listed in
Table 1, have been simulated to get the electric and
magnetic fields both on axis and 5 mm off axis. Using the
transverse electric and magnetic fields, the transverse kick
could be calculated according to [8] as
∫ (
)
(1a)
∫ (
)
(1b)
Meanwhile, using the longitudinal electric field off
axis, the transverse field could be also calculated using
Panofsky’s theorem [6] as
∫
( )
(2a)
∫
(
)
(2b)
Both methods have been used to cross-check the
calculated transverse kick values. On the other hand, the
model was meshed with various mesh sizes to check the
dependence of the calculated kick values on the mesh
size. Complex kick values corresponding to the
converged fine mesh, we have used, are reported in Table
1.
Displacing the beam pipe by 1 mm in y direction,
would induce a kick of 35-j45 keV in the same direction,
while displacing the spoke in y direction seems to have
more pronounced effect as the kick value would be 83-j2
keV. On the other hand, displacing the spoke in x
direction has a little bit different kick effect, rather than in
y direction, of about 102-j1 keV, which is expected
because of the asymmetry of the cavity geometry. Finally,
displacing the spoke in longitudinal direction has very
negligible kick effect.
Table 1: Transverse Kick Due to Various Geometrical
Variations (in keV for 2 MeV energy gain per cavity).
Margin Using Transverse
Fields
Using Panofsky
Theorem
Beam Pipe
1mm Shift in y
ΔPy.c=35.16-j45.17 ΔPy.c=35.26-j45.29
Spoke 1 mm
Shift in x
ΔPx.c=83.33-j1.8 ΔPx.c=83.96-j1.86
Spoke 1 mm
Shift in y
ΔPy.c=101.67-j1.26 ΔPy.c=101.8-j1.36
Spoke 1 mm
Shift in z
~0 ~0
In fact, the actual value of kick depends on the particle
phase relative to the RF. As, a worst case scenario the
largest transverse kick that could happen is about 102 keV
in y direction with a particle of zero phase, in case of a
spoke shift of 1 mm in the same direction.
Assuming a particle with energy of 10 MeV and
β=0.146 (thus with momentum = 137 MeV), the
calculated maximum kick value would induce a beam
deviation:
(3)
of 0.74 mrad, which could be easily corrected with the
Project X specified correctors.
MEASUREMENTS ON THE FABRICATED
CAVITIES
Six cavities have been already fabricated and were
tested (SSR1-105 to SSR1-110).
CMM Measurements
Upon passing the visual inspection of the cavities, they
were subject to physical dimension measurement using
coordinate measuring machines (CMM). Figure 3 shows
the measuring setup of the CMM. Various dimensions are
checked with special consideration given to the entities
nearby the beam axis, where dimensions M5 and M6
shown in Figure 3 were measured at various locations
along the beam axis; specifically at the beam pipe flange
“out”, half way of the beam pipe “mid” and at the start
point of the cavity gap “in”. Also these measurements
were done at both ends of the spoke; “spk1”, “spk2”.
Table 2 summarizes the various CMM measurements
that were done to check the alignment of the beam pipe
and the spoke with respect to the beam axis in the two
transverse directions; parallel (x) and perpendicular (y) to
the spoke. From these measurements, it seems that the
largest y-misalignment happens in SSR1-109 (0.78 mm,
0.75 mm on the spoke), while the largest x-misalignment
happens in SSR1-108 (0.43 mm, 0.40 mm on the spoke).
Bead Pull Measurements
Off axis bead pull measurements were done on the six
cavities measuring the longitudinal electric field on a 5
mm off axis distance in directions parallel and
perpendicular to the spoke, as shown in Figure 4.
Figure 3: CMM Measurements of SSR1 Cavities.
Table 2: Summary of the CMM Measurements of the
SSR1 Cavities
M5
out
M5
mid
M5
in Spk1 Spk2
M6
in
M6
mid
M6
out
106
y 0 0.13 0.25 0.35 0.23 -0.14 -0.07 0
x 0 0.06 0.12 -0.11 -0.07 0.28 0.02 0
107
y 0 0.01 0.02 -0.22 -0.04 -0.09 -0.05 0
x 0 -0.06 -0.12 -0.09 -0.07 0.056 0.027 0
108
y 0 0.09 0.18 -0.13 -0.26 -0.08 -0.04 0
x 0 0.02 0.38 0.43 0.40 0.3 0.15 0
109
y 0 -0.14 -0.27 -0.78 -0.75 -0.18 -0.09 0
x 0.01 0.15 0.28 -0.41 -0.43 0.01 0 0
110
y 0 -0.10 -0.20 -0.49 -0.31 -0.22 -0.11 0
x 0 0 0 -0.29 -0.24 0.16 0.08 0
Again using Panofsky’s theorem we can calculate the
transverse kick from the measured longitudinal fields.
Alignment of the bead was kept within ± 0.5 mm using a
cross-holed flange, as shown in Figure 4.
Figure 4: Bead-pull measurements of SSR1 cavities.
Table 3: Summary of the Measured Kick of the Different
SSR1 Cavities in keV for 2 MeV energy gain per cavity.
ΔPx.c ΔPy.c
105 -36.6 -15.2
106 3.5 24
107 -25.6 -1.3
108 105.7 3.2
109 -40.1 -154.3
110 -0.63 -72.6
Each measurement was repeated five times and average
real values of these measurements are reported in Table 3
(only the real values are reported, as the imaginary part is
more susceptible to noise).
The largest transverse kick in y direction is of 154 keV
and it occurs in SSR1-109, while the largest one in x
direction is of 106 keV and it occurs in SSR1-108. This is
consistent with the CMM measurements shown
previously in Table 2. A 154 keV kick would induce 1.12
mrad beam deviation, which is still manageable to correct
using the 10 mrad specified correctors of Project X.
CONCLUSION
Transverse kick that could happen in SSR1 cavities due
to geometrical variations of the fabricated cavities from
the designed geometry has been analysed and evaluated.
From fabrication experience, three kinds of variations
were under investigation concerning the alignment of
both the beam pipe and spoke with respect to the beam
axis. Simulation study has been carried out implementing
these variations in the simulation model. CMM
measurements of five fabricated SSR1 cavities were
carried out to investigate the amount of physical
misalignments of the beam pipe and spoke. Bead-pull
measurements were also conducted to evaluate the
transverse kick values in the fabricated cavities.
Simulation and measurements are relatively in good
agreement. Maximum kick in the fabricated cavities is
within 154 keV that would induce about 1.12 mrad beam
deviation, which could be definitely corrected with the 10
mrad specified correctors of Project X.
ACKNOWLEDGMENT
We acknowledge the help of Ted Beale, and Oscar Lira
of Fermilab in doing the CMM measurements.
REFERENCES
[1] Project X: Accelerator Overview,
http://projectx.fnal.gov/.
[2] Giorgio Apollinari, “Project X as a Way to Intensity
Frontier Physics,” Proceedings of Hadron Beam
2008, Nashville, Tennessee, USA.
[3] S. D. Holmes, S. D. Henderson, R. Kephart, J. Kerby,
S. Mishra, S. Nagaitsev, R. Tschirhart, “Project X
Functional Requirements Specification,” Proceedings
of PAC’11, New York, NY, USA.
[4] S. Nagaitsev, “Project X – A new Multi-megawatt
Proton Source at Fermilab,” Proceedings of PAC’11,
New York, NY, USA.
[5] Functional Requirements Specification of PXIE
[6] V. Lebedev, “Focusing in PXIE Cryomodules,”
Projectx database.
[7] Comsol Multiphysics v4.2 user guide.
[8] B. V. Zotter and S. A. Kheifets, “Impedances and
Wakes in High-Energy Particle Accelerators,” World
Scientific Publications, 1998.
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